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Theorem funbrfv2b 6966
Description: Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
Assertion
Ref Expression
funbrfv2b (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵)))

Proof of Theorem funbrfv2b
StepHypRef Expression
1 funrel 6585 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 releldm 5958 . . . . 5 ((Rel 𝐹𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹)
32ex 412 . . . 4 (Rel 𝐹 → (𝐴𝐹𝐵𝐴 ∈ dom 𝐹))
41, 3syl 17 . . 3 (Fun 𝐹 → (𝐴𝐹𝐵𝐴 ∈ dom 𝐹))
54pm4.71rd 562 . 2 (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹𝐴𝐹𝐵)))
6 funbrfvb 6962 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
76pm5.32da 579 . 2 (Fun 𝐹 → ((𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵) ↔ (𝐴 ∈ dom 𝐹𝐴𝐹𝐵)))
85, 7bitr4d 282 1 (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹𝐴) = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106   class class class wbr 5148  dom cdm 5689  Rel wrel 5694  Fun wfun 6557  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  brtpos2  8256  mpocurryd  8293  xpcomco  9101  fseqenlem2  10063  fpwwe2  10681  joinfval  18431  joinfval2  18432  meetfval  18445  meetfval2  18446  tayl0  26418  ofpreima  32682  funcnvmpt  32684  curf  37585  uncf  37586  curunc  37589  fperdvper  45875
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